Transcript
Cold Denaturation in Proteins.
Olivier Collet
LPM, Nancy-Universite
CompPhys07Leipzig, Nov. 2007
Olivier Collet ( LPM, Nancy-Universit e) Cold Denaturation in Proteins. CompPhys07 Leipzig, Nov. 2007 1 / 19
Plan
1 Introduction.
2 The Problem.
3 Model.
4 Results 1. : Two-states phase diagram.
5 Results 2 : Four-states phase diagram.
6 Conclusion.
Olivier Collet ( LPM, Nancy-Universit e) Cold Denaturation in Proteins. CompPhys07 Leipzig, Nov. 2007 2 / 19
Introduction.
Biochemistry
Amino-acids and Bond between amino acids.
Olivier Collet ( LPM, Nancy-Universit e) Cold Denaturation in Proteins. CompPhys07 Leipzig, Nov. 2007 3 / 19
Introduction.
Biochemistry
Proteins are large and linear chains made of amino acids
CH2
OH
−NH−C−CO=
H
CH3
−NH−C−CO=
H
Olivier Collet ( LPM, Nancy-Universit e) Cold Denaturation in Proteins. CompPhys07 Leipzig, Nov. 2007 4 / 19
Introduction.
Biochemistry
Put into water proteins fold in a unique compact structure
CH2
OH
−NH−C−CO=
H
CH3
−NH−C−CO=
H
Water
Hydrophobic
Hydrophilic
Main folding force is the Temperature Dependent Hydrophobic Effecta.
aKauzmann 1959, Balwin 1987, Pratt-Pohorille 2002
Olivier Collet ( LPM, Nancy-Universit e) Cold Denaturation in Proteins. CompPhys07 Leipzig, Nov. 2007 5 / 19
The Problem.
Warm and Cold Denaturations 1.
1Privalov, 1989Olivier Collet ( LPM, Nancy-Universit e) Cold Denaturation in Proteins. CompPhys07 Leipzig, Nov. 2007 6 / 19
Model.
Statistical Physics approach.
Hmic = E (m)intr + E (mm′)
solv
m : protein conformationm′ : water configuration
⇒ Z (T ) =ΩX
m=1
Ω′(m)X
m′=1
exp
−E (m)
intr + E (mm′)solv
T
!
Ω′(m)X
m′=1
exp
−E (mm′)
solv
T
!
= z(m)solv(T ) = exp
−F (m)
solv(T )
T
!
⇒ Z (T ) =ΩX
m=1
exp
−H
(m)eff (T )
T
!
with H(m)eff (T ) = E (m)
intr + F (m)solv(T )
Olivier Collet ( LPM, Nancy-Universit e) Cold Denaturation in Proteins. CompPhys07 Leipzig, Nov. 2007 7 / 19
Model.
Effective hamiltonian.
first shell solvent cell
neat solvent cell
solvent nod
monomer
∆(m)ij = 1 if i and j are first neighbors.
Bij : coupling between i and jn(m)
i : number of cells arround ifi(T ): free energy of this solvent cell.n(m)
s : total number of solvent cellsfs(Bs, T ) : free energy of neat solvent
H(m)eff =
X
i>j+1
Bij ∆(m)ij
| z
previous works
+X
i
n(m)i fi(T ) + 2n(m)
s fs(Bs, T )
| z
Collet 2001;2005
More complicated form than the usual : H(m)eff =
X
i>j+1
Bij ∆(m)ij !
Constant of the model.
total lattice links :X
i
X
j
12
∆(m)ij +
X
i
n(m)i + n(m)
s = K1
links of monomeri :X
j
∆(m)ij + n(m)
i = 4Olivier Collet ( LPM, Nancy-Universit e) Cold Denaturation in Proteins. CompPhys07 Leipzig, Nov. 2007 8 / 19
Model.
Effective hamiltonian.
first shell solvent cell
neat solvent cell
solvent nod
monomer
∆(m)ij = 1 if i and j are first neighbors.
Bij : coupling between i and jn(m)
i : number of cells arround ifi(T ): free energy of this solvent cell.n(m)
s : total number of solvent cellsfs(Bs, T ) : free energy of neat solvent
H(m)eff =
X
i>j+1
Bij ∆(m)ij
| z
previous works
+X
i
n(m)i fi(T ) + 2n(m)
s fs(Bs, T )
| z
Collet 2001;2005
More complicated form than the usual : H(m)eff =
X
i>j+1
Bij ∆(m)ij !
Constant of the model.
total lattice links :X
i
X
j
12
∆(m)ij +
X
i
n(m)i + n(m)
s = K1
links of monomeri :X
j
∆(m)ij + n(m)
i = 4Olivier Collet ( LPM, Nancy-Universit e) Cold Denaturation in Proteins. CompPhys07 Leipzig, Nov. 2007 8 / 19
Model.
Effective hamiltonian.
(
n(m)i = 4 −
P
j ∆(m)ij
n(m)s = 1
2
P
i
P
j ∆(m)ij + K1 − 4N
H(m)eff =
X
i>j+1
Bij ∆(m)ij +
X
i
n(m)i fi(T ) + 2n(m)
s fs(Bs, T )
Effective Couplings.
H(m)eff (Bs, T ) =
X
i
X
j>i
Beffij (Bs, T ) ∆
(m)ij
with Beffij (Bs, T ) = Bij − fi(T ) − fj(T ) + 2fs(Bs, T )
- takes a simple form- which forms for fi(T ) and fs(Bs, T ) ?
Olivier Collet ( LPM, Nancy-Universit e) Cold Denaturation in Proteins. CompPhys07 Leipzig, Nov. 2007 9 / 19
Model.
Effective hamiltonian.
(
n(m)i = 4 −
P
j ∆(m)ij
n(m)s = 1
2
P
i
P
j ∆(m)ij + K1 − 4N
H(m)eff =
X
i>j+1
Bij ∆(m)ij +
X
i
n(m)i fi(T ) + 2n(m)
s fs(Bs, T )
Effective Couplings.
H(m)eff (Bs, T ) =
X
i
X
j>i
Beffij (Bs, T ) ∆
(m)ij
with Beffij (Bs, T ) = Bij − fi(T ) − fj(T ) + 2fs(Bs, T )
- takes a simple form- which forms for fi(T ) and fs(Bs, T ) ?
Olivier Collet ( LPM, Nancy-Universit e) Cold Denaturation in Proteins. CompPhys07 Leipzig, Nov. 2007 9 / 19
Model.
Solvation Model
fs(Bs, T ) = Bs − αT ln Ns
Small Bs ⇒ bad solventLarge Bs ⇒ good solvent
n(Bi) =2Ns exp
−B2
i2σ
2
!
σ
√2π erfc
Bmini
σ
√2
!
zi(T ) =
Z ∞
Bmini
n(Bi) exp„
−Bi
T
«
dBi
fi(T ) = −σ
2
2T− T ln
0
B@Ns
erfc“
Bmini
σ
√2− σ
√2
2T
”
erfc“
Bmini
σ
√2
”
1
CA
Olivier Collet ( LPM, Nancy-Universit e) Cold Denaturation in Proteins. CompPhys07 Leipzig, Nov. 2007 10 / 19
Model.
Solvation Model
fs(Bs, T ) = Bs − αT ln Ns
Small Bs ⇒ bad solventLarge Bs ⇒ good solvent
n(Bi) =2Ns exp
−B2
i2σ
2
!
σ
√2π erfc
Bmini
σ
√2
!
zi(T ) =
Z ∞
Bmini
n(Bi) exp„
−Bi
T
«
dBi
fi(T ) = −σ
2
2T− T ln
0
B@Ns
erfc“
Bmini
σ
√2− σ
√2
2T
”
erfc“
Bmini
σ
√2
”
1
CA
Olivier Collet ( LPM, Nancy-Universit e) Cold Denaturation in Proteins. CompPhys07 Leipzig, Nov. 2007 10 / 19
Model.
Solvation Model
fs(Bs, T ) = Bs − αT ln Ns
Small Bs ⇒ bad solventLarge Bs ⇒ good solvent
n(Bi) =2Ns exp
−B2
i2σ
2
!
σ
√2π erfc
Bmini
σ
√2
!
zi(T ) =
Z ∞
Bmini
n(Bi) exp„
−Bi
T
«
dBi
fi(T ) = −σ
2
2T− T ln
0
B@Ns
erfc“
Bmini
σ
√2− σ
√2
2T
”
erfc“
Bmini
σ
√2
”
1
CA
Olivier Collet ( LPM, Nancy-Universit e) Cold Denaturation in Proteins. CompPhys07 Leipzig, Nov. 2007 10 / 19
Model.
The Chain and the Statistical averages.
The chain.
16-mersin 2D lattice
802 075 conformations116 579 extended conformations69 more maximally compact conf
Nat
Statistical average 〈X (T )〉 =PΩ
m=1 X (m)(T )P(m)eq (T ) with
P(m)eq (T ) = 1
Z(T )exp
„
−H(m)
effT
«
.
Compactness: 〈Nc(Bs, T )〉 where N(m)c = 1
9
PNi>j ∆
(m)ij
Order parameter: 〈Q(Bs, T )〉 where Q(m) = 19
PNi>j ∆
(m)ij ∆Nat
ij
Chain entropy, Sch: −〈ln P(m)eq (Bs, T )〉
Olivier Collet ( LPM, Nancy-Universit e) Cold Denaturation in Proteins. CompPhys07 Leipzig, Nov. 2007 11 / 19
Model.
The Chain and the Statistical averages.
The chain.
16-mersin 2D lattice
802 075 conformations116 579 extended conformations69 more maximally compact conf
Nat
Statistical average 〈X (T )〉 =PΩ
m=1 X (m)(T )P(m)eq (T ) with
P(m)eq (T ) = 1
Z(T )exp
„
−H(m)
effT
«
.
Compactness: 〈Nc(Bs, T )〉 where N(m)c = 1
9
PNi>j ∆
(m)ij
Order parameter: 〈Q(Bs, T )〉 where Q(m) = 19
PNi>j ∆
(m)ij ∆Nat
ij
Chain entropy, Sch: −〈ln P(m)eq (Bs, T )〉
Olivier Collet ( LPM, Nancy-Universit e) Cold Denaturation in Proteins. CompPhys07 Leipzig, Nov. 2007 11 / 19
Results 1. : Two-states phase diagram.
Results 1. : Two-states phase diagram
Olivier Collet ( LPM, Nancy-Universit e) Cold Denaturation in Proteins. CompPhys07 Leipzig, Nov. 2007 12 / 19
Results 1. : Two-states phase diagram.
Results 1. : Two-states phase diagram
Olivier Collet ( LPM, Nancy-Universit e) Cold Denaturation in Proteins. CompPhys07 Leipzig, Nov. 2007 13 / 19
Results 2 : Four-states phase diagram.
Results 2 : Four-states phase diagram.
α = 0.5 → 0.9
-15
-10
-5
1 2 34 5
0.0
0.5
1.0
B
T
Q
s
< >
-15
-10
-5
1 2 34 5
0.0
0.5
1.0
B
T
N
s
c< >
-15
-10
-5
1 2 34 5
0
5
10
15
B
T
S
s
ch
Olivier Collet ( LPM, Nancy-Universit e) Cold Denaturation in Proteins. CompPhys07 Leipzig, Nov. 2007 14 / 19
Results 2 : Four-states phase diagram.
Results 2 : Four-states phase diagram.
0 1 2 3 4 5T
0.00
0.01<
Nc>
Bs=0Bs=2Bs=4Bs=6
0 1 2 3 4 5T
11.70
12.00
S ch
Bs=0Bs=2Bs=4Bs=6
0 1 2 3 4 5T
0.0
0.1
0.2
0.3
0.4
0.5
TdS
ch/d
T
Bs=0Bs=2Bs=4Bs=6
Olivier Collet ( LPM, Nancy-Universit e) Cold Denaturation in Proteins. CompPhys07 Leipzig, Nov. 2007 15 / 19
Results 2 : Four-states phase diagram.
Result 2 : Four-states phase diagram.
Olivier Collet ( LPM, Nancy-Universit e) Cold Denaturation in Proteins. CompPhys07 Leipzig, Nov. 2007 16 / 19
Conclusion.
Conclusions.
Calculation of H(m)eff (T ) also simple than E (m)
intr
Cold Denaturation due to hydrophobic effect
Realistic couplings must
take into account of the temperature dependenceof the hydrophobic effect.
Olivier Collet ( LPM, Nancy-Universit e) Cold Denaturation in Proteins. CompPhys07 Leipzig, Nov. 2007 17 / 19
Conclusion.
Conclusions.
Calculation of H(m)eff (T ) also simple than E (m)
intr
Cold Denaturation due to hydrophobic effect
Realistic couplings must
take into account of the temperature dependenceof the hydrophobic effect.
Olivier Collet ( LPM, Nancy-Universit e) Cold Denaturation in Proteins. CompPhys07 Leipzig, Nov. 2007 17 / 19
Conclusion.
Olivier Collet ( LPM, Nancy-Universit e) Cold Denaturation in Proteins. CompPhys07 Leipzig, Nov. 2007 18 / 19
Conclusion.
Effective Couplings as function of T .
0,2 0,4 0,6 0,8T
-25
-20
-15
free
ene
rgy
fi+f
j
Bij+2f
s ; B
s=-4.0
Bij+2f
s ; B
s=-5.5
Bij+2f
s ; B
s=-6.0
Bij+2f
s ; B
s=-7.0
Tij
Tij
-
+
(Bs=-6.0)
(Bs=-6.0)
Figure: Curves of the different contributions to the effective coupling between the monomers 1and 4 as function of the temperature for several values of the solvent quality. The twotemperatures for which the coupling vanishes are shown for Bs = −6.0.
Beffij (Bs, T ) = Bij − fi(T ) − fj(T ) + 2fs(Bs, T ) may be :
positive at low Tnegative at medium Tpositive at high T
Olivier Collet ( LPM, Nancy-Universit e) Cold Denaturation in Proteins. CompPhys07 Leipzig, Nov. 2007 19 / 19
Conclusion.
Effective Couplings as function of T .
0,2 0,4 0,6 0,8T
-25
-20
-15
free
ene
rgy
fi+f
j
Bij+2f
s ; B
s=-4.0
Bij+2f
s ; B
s=-5.5
Bij+2f
s ; B
s=-6.0
Bij+2f
s ; B
s=-7.0
Tij
Tij
-
+
(Bs=-6.0)
(Bs=-6.0)
Figure: Curves of the different contributions to the effective coupling between the monomers 1and 4 as function of the temperature for several values of the solvent quality. The twotemperatures for which the coupling vanishes are shown for Bs = −6.0.
Beffij (Bs, T ) = Bij − fi(T ) − fj(T ) + 2fs(Bs, T ) may be :
positive at low Tnegative at medium Tpositive at high T
Olivier Collet ( LPM, Nancy-Universit e) Cold Denaturation in Proteins. CompPhys07 Leipzig, Nov. 2007 19 / 19
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